interface traps e-h puddles gfet 2011

8/6/2019 Interface Traps E-H Puddles GFET 2011

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INFLUENCE OF INTERFACE TRAPS AND ELECTRON-HOLE PUDDLES ON QUANTUM

CAPACITANCE AND CONDUCTIVITY IN GRAPHENE FIELD-EFFECT TRANSISTORS

G. I. Zebrev, E.V. Melnik, A.A. Tselykovskiy

Department of Micro- and Nanoelectronics, National Research Nuclear University MEPHI, 115409,

Kashirskoe sh., 31, Moscow, Russia

Abstract

We study theoretically an influence of the near-interfacial insulator traps and electron-

hole puddles on the small-signal capacitance and conductance characteristics of thegated graphene structures. Based on the self-consistent electrostatic consideration andtaking into account the interface trap capacitance the explicit analytic expressions forcharge carrier density and the quantum capacitance as functions of the gate voltagewere obtained. This allows to extract the interface trap capacitance and density ofinterface states from the gate capacitance measurements. It has shown that self-

consistent account of the interface trap capacitance enables to reconcile discrepanciesin universal quantum capacitance vs the Fermi energy extracted for different samples.

The electron-hole puddles and the interface traps impact on transfer I-V characteristicsand conductivity has been investigated. It has been shown that variety of widths ofresistivity peaks in various samples could be explained by different interface trapcapacitance values.

1. Introduction

Capacitance measurements provide important information about density of states of mobile and localized

states at the Fermi energy in the 2D and quasi-2D systems. The former is connected with the electronic

compressibility and is often referred to as quantum capacitance [1]. The latter is associated with the interface

traps which are capable to change their occupancy with gate bias changes and have energy levels distributed

throughout the insulator bandgap [2].The concept of quantum capacitance was introduced by Luryi [1] in order to develop an equivalent circuit

model for devices that incorporate a highly conducting two-dimensional (2D) electron gas. The quantum

capacitance can be considered as a direct generalization of the inversion layer capacitance in the silicon

MOSFETs to the case of strictly one-subband filling. The inversion layer (quantum) capacitance plays rather

minor role in the silicon FETs since it is very low in subthreshold operation mode and extremely high in above

threshold strong inversion regime. In the former case the quantum capacitance in MOSFETs is masked by the

parasitic interface trap and depletion layer capacitances connected in parallel in the equivalent electric circuit,

and in the latter case it is insignificant due to the series connection with the gate insulator having typically lesser

capacitances for high carrier densities in inversion layers. In fact the inversion layer capacitance in MOSFETs is

only important in the very narrow region of weak inversion where it is commensurable with the oxide anddepletion layer capacitances. Graphene based FETs bring in absolutely new state of affairs. Quantum

capacitance in graphene has an absolute minimum at the charge neutrality point, which in itself is not so small at

room temperatures even for ideal graphene ( 10fF/m2 at 300 K) and slowly increases as linear function of theFermi energy. In all range of the Fermi energy it may be commensurable with the capacitance of parasitic

interface traps which unavoidable occur at the interface due to chemical and/or structural disorder. The

depletion layer is absent in GFETs and the interface traps capacitance is the only in parallel connection with the

quantum capacitance in the equivalent electric circuit. It is well known that high density of interface traps

suppress electric field effect in gated structures generally and degrades field-effect mobility in particular [3].

Therefore the role of fast interface traps in operation of graphene gated structure as a FET needs to be

understood [4]. This leads to importance of experimental discernment of interface traps and quantum

() [email protected]


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capacitances since the interface trap density (and capacitance) can vary in a wide range of values depending on

purity and quality of the interface or the substrate.

Interface traps act in similar way on transfer I-V characteristics of graphene FETs. Distorting the dependence

of the gate voltage on the chemical potential the interface trap buildup leads to reduction of transconductance

(or, field-effect mobility) even at permanent scattering rate or true mobility. Electron-hole puddles in graphene

are another consequence of presence of the near-interfacial charged defects need not to be easily rechargeable.

Electron-hole puddles modify quantum capacitance and conductivity near the charge neutrality point increasing

its minimum value. The observed minimums of small-signal C-V characteristics are also strongly influenced by

electron-hole puddles. The aim of this work is to develop a regular procedure for separation of the interface and

quantum capacitances based on experimental capacitance data and to investigate the influence of the interface

traps and the electron-hole puddles on capacitance and conductivity characteristics of graphene field-effect

transistors.

The paper is organizes as follows. Sec.2 is devoted to derivation of the expression for quantum capacitance in

ideal graphene without disorder and interaction. The interface traps are briefly discussed in Sec.3. General

electrostatics and specific electrostatic parameters are the topics of Sec.3 and 4. Analytic expressions for

graphene sheet charge density vs gate voltage and channel and gate capacitances taking into account interface

trap density have derived in Sec.6 and 7. Analysis of literature experimental data and a separation procedure are

presented in Sec.8 and 9. Impact of electron-hole puddles on quantum capacitance and mean conductivity ofinhomogeneous graphene is considered in Sec.10 and 11.

2. Quantum Capacitance in Graphene

The density of states in clean graphene for dispersion law 2 20 x yv p p= + is given by

( )2 2 2 2 20 0

22sgnDg

v v

= =

, (1)

where is the Plank constant, v0 ( 108 cm/s) is the characteristic (Fermi) velocity in graphene.

Using the equilibrium Fermi-Dirac functionfFD() the electron density per unit area ne at a given chemical

potential for nonzero temperature Treads

( ) ( ) ( )

( )

20

22

22 2 00 0

2 2

1 exp

B

e D FD

B k TB

B

n d g f

k T u k Tdu Li e

v vu

k T

+

+

= =

= = +

, (2)

where T is absolute temperature, kB is the Boltzmann constant,Lin(x) is the poly-logarithm function of n-th order[5]

( ) 1k n

n k Li z z k

== . (3)Using electron-hole symmetryg()=g() we have similar relationship for the hole density nh

( ) ( ) ( )( ) ( )2

0

2 2

0

21 B

k TBh D FD e

k Tn d g f Li e n

v

= = =

. (4)

Full charge density per unit area or the charge imbalance reads as

( ) ( ) ( )( )2

2 20

0

2.B B

k T k T BS e h

k Tn n n d g f f Li e Li e

v

+ = + =

(5)

Conductivity of graphene charged sheet is determined by the total carrier density

2

2 2

0

2B Bk T k T B

S e h

k T N n n Li e Li e

v

= + = +

, (6)


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notice that ( ) ( )( )2

00 / 3S B N k T v = = .

The channel electron density per unit area for degenerate system ( >>kBT) reads

( )2

2 2 200

S Dn d gv

=

. (7)

Performing explicit differentiation of Eqs.(2,4) one reads

2 2

0

2 ln 1 expe BB

dn k T d v k T

= +

, 2 20

2 ln 1 exph BB

dn k T d v k T

= +

. (8)

Exact expression for quantum capacitance of the graphene charge sheet may be defined as

( )( ) 20

0 0

2ln 2 2cosh .

e h BQ

B

e d n nf k TeC g d

d v v k T

+

= = +

(9)

3. Near-interfacial rechargeable oxide traps

It is widely known (particularly, from silicon-based CMOS practice) that the charged oxide defects inevitably

occur nearby the interface between the insulated layers and the device channel. Near-interfacial traps (defects)

are located exactly at the interface or in the oxide typically within 1-3 nm from the interface. These defects canhave generally different charge states and capable to be recharged by exchanging carriers (electrons and holes)

with the device channels. Due to tunneling exchange possibility the near-interfacial traps sense the Fermi level

position in graphene. These rechargeable traps tend to empty if their level t are above the Fermi level and

capture electrons if their level are lower the Fermi level.

EF

VNP

VG >VNP

EF

t t

VG < VNP

(a)

(a) (b)

Fig. 1. Illustration of carrier exchange between graphene and oxide defects (a) filling; (b) emptying.

There are two types of traps donors and acceptors. Acceptor-like traps are negatively charged in a filled state

and neutral while empty ( - /0). Donor-like traps are positively charged in empty state and neutral in filled

condition (0/+). In any case, the Fermi level goes down with an increase VG and the traps begin filled up, i.e.

traps become more negatively charged (see Fig. 1). Each gate voltage corresponds to the respective position of

the Fermi level at the interface with own equilibrium filling and with the respective density of equilibrium

trapped charge Qt() = eNt() which is assumed to be positive for definiteness. For traps with small recharging

time the equilibrium with the substrate would establish faster. These traps rapidly exchanged with the substrate

are often referred as to the interface traps (Nit) [6], [7]. Defects which do not have time to exchange charge with

the substrate during the measurement time (gate bias sweeping time) are referred to as oxide-trapped traps (Not).

Difference between the interface and oxide traps is relative and depends, particularly, on the gate voltage sweep

rate and the measurements temperature. Interface trap capacitance per unit area Cit may be defined in a

following way

( )( ) 0it td

C eNd

> . (10)

Note that the Fermi level dependent eNt() contains the charge on all traps, but for a finite voltage sweep time tsonly the interface traps with low recharging time constants t < ts contribute to the recharging process.


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0

gate gate

gate graphene ox ox ox

ox ox

eN eN E d d

C

= = , (17)

where Ngate(VG) is the number of charge carriers on the metallic gate per unit area and the oxide (insulator)

capacitance per unit area Cox expressed through the dielectric constants of the insulator (ox) is defined as

0oxox

ox

Cd

= . (18)

5. Characteristic Scales of Gated GrapheneThe planar electric charge neutrality condition for the total gated structure can be written down as follows

G t S N N n+ = , (19)

whereNG is the number of positive charges per unit area on the gate; ns is the charge imbalance density per unitarea (ns may be positive or negative and generally non-integer), Nt is the defect density per unit area which isassumed to be positively charged (see Fig.3). Then total voltage drop (Eq.16) across the structure becomesmodified as

( ) ( )( )

2

G gg S t ox

e

eV n N C = + + . (20)

dox

GATE

OXIDE

GRAPHENESHEET

VG =e

Nt

Fig. 3. Band diagram of graphene FET.

The voltage corresponding the electric charge neutrality point gate VNPis defined in a natural way

( )

( )0

0

t

NP G gg ox

eN

V V C

=

= = . (21)

Chemical potential is positive (negative) at VG > VNP(VG < VNP). Then we have

( )( ) ( )( )22 0t tS

G NP

ox ox

e N Ne ne V V

C C

= = + + . (22)

Taking for brevity without loss of generality VNP =0 and assuming zero interface trap charge at the NP point

as well as constant density of trap states we have

( ) ( )( )2

0t t it e N N C = . (23)

Taking into account Eq.23 the basic equation of graphene planar electrostatics can be written down in a form


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2 2

2

S it F G F F F

ox ox a

e n CeV m

C C

= + + + , (24)

where we have introduced for convenience a dimensionless ideality factor

1 it

ox

Cm

C + , (25)

and notation F used instead of . The specificity of the graphene-insulator-gate structure electrostatics is

reflected in Eq.24 in appearance of the characteristic energy scale2 2

0 0

22 8

ox oxa

G ox

v C v

e d

= =

, (26)

where the graphene fine structure constant is defined as ( in SI units)

2

0 04

G

e

v

=

. (27)

This energy is nothing but the full electrostatic energy stored in the capacitor with the area fall at one carrier in

graphene.

Fig.4 shows dependencies of characteristic electrostatic energy of gated graphene a vs gate oxide thickness

for typical dielectric constants 4 (SiO2) and 16 (HfO2).

1 2 5 10 20 50 100 200

0.001

0.005

0.010

0.050

0.100

0.500

1.000

dox, nm

ea,eV

Fig. 4. The dependencies of the a as functions of the insulator thickness dox for different dielectric permittivity equal to 4 (lower

curve) and 16 (upper curve).

Energy scale a bring in a natural spatial scale specific to the graphene gated structures2

0

0

2 8 GQ ox

a ox ox

v ea d

v C

= =

, (28)

and corresponding characteristic density

( )2

2 2 2

0

1 aQ S F a

Q

n na v

= = =

. (29)

Due to the fact that graphene fine structure constant G 2.0 2.2 the characteristic length aQ is occasionallyof order of the oxide thickness for the insulators with ox ~ 16 (i.e. for HfO2). Interestingly that the energy scale

a can be as well represented as functions of the Fermi energy and wavevector kF, quantum capacitance andcharge density

2

1 1a ox

F Q F Q S Q

C

C k a n a

= = = , (30)

where is defined independently as the ratio of the diffusion to the drift component in the channel [9].


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6. Self-Consistent Solution of Basic Electrostatic Equation

Solving algebraic Eq. (24) one obtains an explicit dependence (to be specific for VG > 0) of the electronFermi energy as function of the gate voltage

( )1/ 2

2 2 2 F a a G am eV m = + (31)

This allows to immediately write the explicit relation for graphene charge density dependence on gate voltage

( )2

1/ 22 2 2 2S G F G a a a G

ox

e neV m eV m m m eV

C = = + + . (32)

Restoring omitted terms the latter equation can be rewritten as [10], [11]

( )1 2

0

0

1 1 2G NP

S G ox G NP

V Ven V C V V V

V

= + + , (33a)

where the characteristic voltage V0 m2a / e is defined where interface trap capacitance is taken into account.

One can modify Eq.33 taking into account the finite total carrier density ( )0SN = at the charge neutralitypoint

( )1 22

0

0 0

1 1 23

G NPBS G ox G NP

V Vk Ten V e C V V V

v V

= + + +

. (33b)

The modified Eq.33b yields results almost identical to the formally exact Eq.6.

Figs. 5-6 exhibit numerically the interrelation ofV0 with Citand dox.

dox=10 nm, eox=5.5

dox=300 nm, eox=4

0.05 0.10 0.50 1.00 5.00

0.01

0.1

1

Cit, fFmm2

V0

,V

Fig. 5. Simulated dependencies of the characteristic voltage V0 as functions of the interface trap capacitance Cit

for different oxide parameters.

Cit = 0

Cit = 2

Cit = 1

Cit = 3

10 1005020 20030 30015 15070

0.005

0.010

0.050

0.100

dox, nm

V0

,V

Fig. 6. Simulated dependencies of the characteristic voltage V0 as functions of oxide thickness

for different interface trap capacitance (in fF/m2).


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Fig. 8. Equivalent circuit of gated graphene.

One might introduce another relation corresponding to the intrinsic channel capacitance

11

QS S oxCH

Q it ox it G G

Qox

C N dN d C C e e

C C C C V dV d

CC

= = = = + + ++

, (38)

where all capacitances are non-zero and assumed to be positive values for any gate voltage. The gate and the

channel capacitances are connected in graphene gated structures through exact relation

1G it

CH Q

C CC C

= + (39)

and can be considered to be coincided only for ideal devices without interface traps when Cit = 0. All

relationships for the differential capacitances remain valid for any form of interface trap energy spectrum. In an

ideal case capacity-voltage characteristics CCH(VG) should be symmetric with refer to the neutrality point

implying approximately flat energy density spectrum of interface traps. For the latter case the channel capacity

can be derived by direct differentiation of explicit dependence nS(VG) in Eq.33

1/ 2

0

11

1 2

SCH ox

G G NP

dnC e C

dV V V V

= = +

. (40)

As can be seen in Fig.9 the capacitance-voltage characteristics CG(VG) is strongly affected by the interfacetrap capacitance.

Fig. 9. Simulated dependencies of the gate capacitance CG(VG) for different Cit= 1, 5, 10fF/m2; dox = 10 nm, ox = 5.5 (Al2O3).

For the case Cit= 0 (i.e. m = 1) capacitance-voltage dependencies can be considered as to be universal curves

depending on only thickness and permittivity of the gate oxide through the parametera. In practice one should

discriminate the quantum and the interface trap capacitances and this is a difficult task since they are in a

parallel connection in equivalent circuit. Comparison of ideal capacitance voltage characteristics with real

measured ones represent a standard method of interface trap spectra parameter extraction [2], [13].


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8. Experimental Data AnalysisCapacitance vs gate voltage dependencies in graphene gated structures have reported by several experimental

groups followed by recalculation of quantum capacitance [14], [15], [16], [17], [18]. Although the

measurements showed the expected V-shape dependence centered at the NP, the data, as a rule, are still far from

ideal, making difficult to extract consistently the parameters of universal quantum capacitance dependence on

the Fermi energy. Particularly, Fig.10 shows quantum capacitance dependencies adapted from independent

results performed by the two independent experimental groups [14], [18].

Fig. 10. Quantum capacitance vs the Fermi energy points recalculated from the capacitance data by the two experimental groups. The

upper set of circles (red online) corresponds to the gate (Cox = 4.7fF/m2 [18]), and lower circles (blue online) represent data of Ref. [14]

(Cox = 5.6fF/m2). These data are described by Eq. 9 with v0 1.15 10

8 cm/s for the upper curve (as obtained in original Ref. [18]) and

v0 1.5 108 cm/s for the lower curve (as computed by us based on data in Ref. [14])

Despite of the both curves seem to be rather symmetric (especially, the upper) but they are obviously not

coincident and admittedly far from a unique universal dependence which has to be described by ideal Eq.9. We

argue here that the pointed disagreement follows from lack of consistent account of the interface trap

capacitance under recalculation from initial measured capacitance data to quantum capacitance. Determination

of the interface trap and quantum capacitances has to be considered as simultaneous and self-consistentprocedure of their separation. Moreover the characteristic graphene velocity could be corrected in certain limits

to adjust recalculated experimental quantum capacitance dependencies to the known universal curve.

9. Quantum and Interface Trap Capacitance Separation Procedure

If we have ideal structure with Cit = 0 (i.e. m =1) then the gate capacitance depends only on a single

dimensional parametera containing the gate oxide capacitance and the characteristic graphene velocity v0. In

practice one should discriminate quantum interface trap capacitance and this is a difficult task since it is in

parallel connection with the interface trap capacitance in an equivalent electric circuit.

A following iteration procedure can be used for separation. At first we set Cit= 0 (m = 1) and

( )1 /G CH it Q CH C C C C C = + (recall that CQmin 10fF / m2

at room temperatures). Then one can replot the

experimental data forCG (VG) using a reformulation of Eq.40

( ) ( )0 0

2 2

1 11 1

2 21 1G NP

CH ox G ox

V VV V

C C C C

=

. (41)

Fig.11 shows a typical result of such graphical representation of experimental data which turns out to be linear

in full agreement with Eq.41.


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0 20 40 60 80 100 120 140

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

H1-CGCoxL-2 - 1

VG,

V

Fig. 11. Replot of the capacitance data illustrating the finding ofCitwith least-squares method. Upper line (blue circles online)

corresponds to data in Ref. [13], the lower (red circles online) line corresponds to data in Ref. [17].

The slope of this linear dependence yields an experimental value of the characteristic voltage V0 whichdepends on Cit. If we were aware exactly v0 and Cox one could immediately to obtain m and Cit. The ratio of the

slopes does not depend on0

v (indices 1 (2) correspond to the data in Ref. [14] (Ref. [18]))

( )( )

20 11

2

0 2 2

1

2

ox

ox

V Cm

V m C= (42)

and can be determined immediately from the Fig.11 V0 (1)/ V0 (2) = 1.237. Using the known oxide capacitances

we have found the ratio m1/m2 = 1.02 and Cit1/ Cit2 = 1.53.

Setting as a zero approximation v0 = 1.15 108

cm/s (found in Ref.[18]) one can compute Cit which turn to be

non-equal but both of order 1 fF/m2 for results of the both experimental groups [14, 18] in contrast to own

values used by the authors. Recall, that the values Cit = 10 fF/m2

[14] and Cit = 0 [18] were used at datatreatment as far as we know. Furthermore with a use Eqs.37 the first iteration for quantum capacitance as

function of Fermi energy can be calculated through experimental data

1

1 1Q it

G ox

C CC C

=

. (43)

Recalculated in this manner experimental points for CQ are found to be lower than original results in [18]

where interface trap capacitance were ignored and to be higher than in [14] where itC were overestimated. In

addition the experimental points from independent data have laid practically on a single curve, corresponding to

characteristic graphene velocity in the range 1.15108 cm/s < v0< 1.5108

cm/s.

Iterating the procedure for self-consistency we have found a following set of best fit parameters representedin Table 1.

Table 1

ReferenceoxC ,fF/m

2 m itC ,fF/m2

0v , 108 cm/s

[Chen et al., 2009] 5.6 [14] 1.10 0.55 1.300.05[Ponomarenko et al., 2010] 4.7 [18] 1.08 0.36 1.300.05

Comparison of original and recalculated by us dependencies shown in Fig.12 exhibits the fact that consistent

account of the interface trap capacitance is a necessary condition for obtaining of universal parameter of

quantum capacitance.


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Fig. 12. Quantum capacitance curves recalculated as functions of the Fermi energy. Upper (purple) curve corresponds to [18] data and

simulation with v0 =1.15108 cm/s (with ideal CQ dependence). The lower (blue circles online) curve corresponds to [14] which is

simulated by us with v0 =1.5108 cm/s (in fact from Fig.8. of [14]). Taking into account extracted interface capacitances the experimental

points of both groups have laid on a single medium (red) curve with v0 =1.3108 cm/s (yellow squares, [18]; green squares [14]).

Notice that found interface trap capacitances correspond numerically to a reasonable range of interface trap

density of statesDit (2.2 3.4) 1011 cm-2 eV-1 typical for pristine gate oxides in the silicon MOSFETs.

Fig. 13a. Comparison of the experimental gate capacitancedependence CG(VG) obtained in [18] (points) and simulation with

the Eqs.37 (blue) and 40 (red) (taking into account Eq.39). Used

constants are v0 =1.3108 cm/s, Cox = 4.7 fF/m

2 [18], Cit(extracted) = 0.36fF/m2. The dashed curve shows Cox.

Fig. 13b. Comparison of the experimental gate capacitancedependence CG(VG) obtained in [14] (points) and simulation with

the Eqs.37 (blue) and 40 (red) (taking into account Eq.39). Used

constants are v0 =1.3108 cm/s, Cox = 5.6 fF/m

2 [14], Cit(extracted) = 0.55fF/m2 (unlike to Cit= 10fF/m

2 used in [14]).

Fig.13 (a and b) shows the experimental gate capacitance as functions of gate voltage obtained in papers [18]

and [14]. Based on formulas for homogeneous graphene the gate capacitance estimations strongly underestimate

the capacitance values nearby the charge neutrality point. As can be seen in Figs.13 the differences between the

experimental and calculated values correlate with the interface trap density: the greater disorder and concerned

with it the interface trap capacitance the greater underestimation for homogeneous graphene approximation. It is

known that due to occurrence of potential fluctuation induced by the charged oxide defects, graphene charge

sheet breaks near the neutrality point into electron and hole puddles [19, 20]. This electron-hole puddles are

capable to significantly increase the minimal quantum capacitance value that may yields independent

information about charged defects density trapped in the insulator near the graphene sheet.


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10. Electron-hole puddles and quantum capacitance

The potential fluctuation induced by charged near-interfacial defects distributed in uncorrelated way in the

insulator can be described by Gaussian distribution function

( )2

22

1exp

22

uP u

uu

=

(44)

where u is fluctuating potential reckoning from a mean value, 2u is the dispersion of potential fluctuation.

The standard deviation for potential of uncorrelated near-interfacial defects can be assessed as [21]

( )

42

2

04

imp

eu n

= (45)

and to be determined by a sum of the positively and negatively charged defect densities( ) ( )

imp imp impn n n+ = + ; is a

half-sum of the permittivities for the dielectrics adjusted to the graphene sheet. In the Thomas-Fermi

approximation the local value of charge density is

( )( ) ( )( )

2

2 2

0

rr FS F

un sgn u

v

=

(46)

where F is a single equilibrium Fermi energy of the inhomogeneous system.

At first we have to calculate the total net electric charge in graphene as function of F taking into account

occurrence of potential fluctuation and electron-hole puddles

( ) ( ) ( ) ( ) ( ) ( )

( )

2 2

2 2

0

22 2 2

2 2 220

2exp .

22

F

F

p

F F F

F FF F

eQ u P u du u P u du

v

eu erf u

v uu

=

= + +

(47)

Then the quantum capacitance accounting the electron-hole puddles becomes

( )

22 2

2 2 220

2 2exp

22

p F F FQ

F F

uQ eC e erf

v uu

= = +

. (48)

The latter relation can be obtained immediately by direct averaging of density of states( ) ( ) ( )2 2 20p

QC e v u P u du

= . The Eqs. 47 and 48 do not contain temperature since to be only valid for a

conditions1/ 2

2

B Fk T u < < or

1/ 22

B Fk T u < < .

At the CNP we have the minimum quantum capacitance in disordered graphene

( ) ( ) ( )22

min 2 2

0

220

p p

Q Q F

ueC C

v

= = =

, (49)

which determines an observed plateau in quantum capacitance dependencies for1/ 2

2

Fu < . Figure 14 shows

the comparison of the experimental and simulated gate capacitance characteristics obtained with correctedquantum capacitance Eq. 48 by fitting of potential standard deviation.


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Fig.14a. Simulated gate capacitance dependence in comparison

with experimental points [17]. Fitted value of potential dispersion

is u21/2=95 meV. The lower curve corresponds to u21/2 = 0.

All other parameters are taken the same as in Fig.13.

Fig.14b. Simulated gate capacitance dependence in comparison

with experimental points [13]. Fitted value of potential dispersion

is u21/2= 141 meV. All other parameters are taken the same as

in Fig.13.

The charged impurity total concentrations nimp computed with Eq.45 are 3.9 1012 cm-2 and 8.6 1012 cm-2 forfitted standard deviations 95 and 141 meV. Recalculated quantum capacitance curves corrected with account ofelectron-hole puddle contribution are depicted in Fig.15. Eq.48 was used in Fig.15 instead of Eq.9 for idealgraphene.

Fig.15. Corrected quantum capacitance curves recalculated for the two experiments taking into account electron-hole puddles. Red (blue)curve corresponds to data taken from [Chen, 2008]([Ponomarenko, 2010]). Lower (green) curve corresponds to the universal relation

(Eq.9) for ideal graphene. All other parameters are taken the same as in Figs. 13-14.

11. Conductivity averaging

The conductivity in graphene is given by the conventional formula 0 0 Se N = , where 0 is the mobility

and SN is the total concentration of carriers of both signs. First, we have calculated the total carrier (electron +

hole) density in inhomogeneous graphene sheet

( ) ( ) ( ) ( )2 2

2

2 2 2 2

0 0

Fp

S F F

ue N u P u du

v v

+ =

. (50)

Obviously, the Eq.50 implies that the residual carrier concentration at the CNP [22] is determined immediatelyby the potential fluctuation dispersion


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( ) ( )2 2

0 22 2

0

0p G

S F imp

ox

un N n

v

= = = =

(51)

This is exactly the result, which can be derived with the Shklovskii argument of nonlinear screening [23] withthe optimal size of the puddles and residual concentration

2

0 2

ox

G imp

Rn

,

2

0

0 2

0

impn Rn

R

. (52)

Similar arguments has been used in Refs. [24] for description of disorder at the Si-SiO 2 interface in the siliconMOSFETs.

To assess the carrier mobility ( )0 0 /F Fev p = the relaxation time can be estimated through the FermiGolden Rule

( )( )

2 2

2 2

1 2 2~

Q

imp imp D F imp imp

tr F

Cn u g n u

e

=

, (53)

where impu is the average of the scattering potential matrix element in 2D momentum space. According to

Ref. [9] the screened matrix element is expressed as

2

imp

Q ox

euC C

=+

, (54)

where the gate screening of the Coulomb scatterers dominates near the CNP at 0 /Q ox ox oxC C d


16/17

16

relation Eq.6 instead of approximate Eq.7 is essentially for smooth description of resistivity near the resistivity

maximum at low temperatures.

The dependences of the Fermi energy F as functions of gate voltage were modeled with Eq.31. Such

computation scheme allows to easily describe the impact of interface trap recharge on the shape of measured

characteristics as functions of external gate voltage. Excepting trivial NPV , we fit only the interface trap

capacitance Cit and the constant residual concentration 0n for three different samples. The total charged defect

density impn then has been recalculated using Eq.51.

Comparison of experimental and simulated results is shown in Fig.16.

Fig. 16.The gate voltage dependence of the resistivity for the samples S1, S2, S3 [24]: the carrier mobilities were taken from [24]

(S1) = 17500 cm

2

/Vs, (S2) = 9300 cm

2

/Vs, and (S3) = 12500 cm

2

/Vs. Fitting results: the sample S1: n0 = 0.671011

cm

-2

, Cit= 3.5fF/m2; the sample S2: n0 = 1.610

11-2, Cit = 9.3 fF/m2; the sample S3: n0 = 0.6410

11-2, Cit = 4.4fF/m2 (T= 200K, ox = 4,

dox = 300 nm, v0= 1.3108 cm/s).

The extracted parameters for different samples are summarized in Table 2.

Table 2

sample Cit,fF/m2 n0, 10

11 cm-2 nimp, 1011 cm-2 0, cm

2/Vs [25]

S1 3.6 0.67 1.5 17500

S2 9.5 1.6 3.5 9300

S3 4.4 0.63 1.4 12500

Simulation results exhibit an excellent agreement with the experiment in description of vicinity of the Diracpeak for resistivity at reasonable values of extrinsic physical parameters. This suggests that the widths of Dirac

peaks are determined mainly by the interface trap density. Behavior of the resistivity dependences at large |VG

VNP| (where 1 k/ ) is typically influenced by the contacts.


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17

12. References

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